There are a few moving parts to make the society desired here. Overall, it's a cascading effect where the super rich and ultra rich will do what they can to preserve their wealth. Some cynics might call this the end result of Capitalism
Note: I have tried to keep this general where I can. Wall of Text ahead
Step 1: Establish a Ruling Dynasty
This is the easy part for societies that have it and a bit more difficult for those that don't. Regardless, this will be a transfer of power that will be highly objected without a global disaster going on or the plutocrats deciding it's a good idea to instill a puppet dynasty and throwing money at the idea.
Regardless of the how, it will be resisted as people do not like to give up their power. In some societies, it would require a change to their constitution (or similar document) to make such a concept even remotely legal in the first place.
Now as to the why we do this: Overall the reason is stability. Something, be it somebody actually blowing up the houses of Parliament/Congress, a pandemic, or war, some trigger event causes enough chaos and mayhem to warrant setting this up. It will be touted as a way to keep the country running, but really be a way for the rich to fully consolidate their power.
For countries that already have a monarch, well this part is technically done. Said monarchs might need to take back some of the power they have given away over the decades.
Step 2: Just in Caste
Well, now you have a ruler with enough power to make the laws. But how to handle the caste system? Well, typing for North America, it's handling itself quite fine already. The rich basically spend money specifically to separate themselves from the poorer folks already. All some people need is a push in a certain direction to land squarely in a caste, and to change the system to keep them there.
Enough people here are essentially the Working Poor that they cannot afford a sick day unless it is a paid one. With rising costs of everything and a paycheck that cannot keep up with those costs, many are stuck poor with no chance of rising out barring a miracle. That is the current reality for many.
Keeping them there is simple in the sense of first never letting them own property. Without the ability to own their own home, they will forever have to pay rent to a higher caste in order to live there. A rent that can be tailored to take enough of their income to keep them poor, but without making homelessness a better alternative. It will be a balancing act.
The second big way to keep them from rising up the caste system is to prevent them from getting any form of loan or financial help that would help them ascend the ladder. This is easy because it will be the Ruling Class that has the money and a simple lookup will show them what class a person belongs to.
The third way is to only allow the Working Class enough education to do the jobs of the class. Most jobs you have listed in for the class only require a high school diploma (or equivalent) to attain with some of those jobs needing some manner of college level education. Put the focus on practical/applied classes and apprenticeships as opposed to classes heavy on theoretical content that they aren't really going to need. It's less keeping them uneducated as much as it is tailoring their education to their needs.
Separation is achieved by physically relocating people to higher density neighbourhoods. Apartment buildings that the Ruling Class owns will be the primary housing here with a handful of small bungalows that the wealthiest of this caste can potentially own with the infrequent duplex/triplex that the Specialist Class owns for extra income.
By offering a very basic healthcare and unemployment funds for this class, it keeps them from doing things that threaten the integrity of the class. The Ruling Class does not want a pandemic to cut through the streets because they can't go to a hospital, and they do not want people in agonizing pain due to disease. But overall, the Working Class is expendable and it is that overall thought that will guide healthcare decisions.
Side note: The big item you are missing for jobs here is Servant. It is not inconceivable for the Ruling Class to have servants or servant families to attend to them.
Specialist jobs require a university degree on average with others needing further education on that, so that covers the education part of it. On that front, only approved schools can offer the degrees needed for Specialists to pursue their jobs. Not only that, there will be less of them in order to centralize them into one easy to track place. They will, of course, be priced such that the Working Class will never be able to afford it unless sponsored by a higher class (for personal retainer purposes likely)
They will never rise to the Ruling Class just because they lack the money to do so, and even if they do accumulate the money, they will never gather the appropriate influence. They are also kept down there by limiting what they own property-wise.
With a base 2020 salary of $86,000 unemployed, there might be some layabouts that cruise through because they are inept, relatively speaking. However, the class is designed to look good, but really keep them just as poor as the Working Class, just a prettier poor.
By not extending basic healthcare to the Specialists, it forces them to either tie up a portion of their money into a savings (held by the Ruling Class' banks) in case something happens, or suffer and be an example to others. By allowing them property in the form of houses, their personal office/stores, and the occasional property to rent, it ties up further income into maintaining those properties. However, they will be limited in how much they can own.
Expect small businesses to be owned by Specialists and have the Working Class work for them.
The Ruling class allows them loans to further deplete their money in the form of interest, and allows them highly limited investments into their ventures, perhaps as a reward for services rendered.
Well, this class already exists at least in part. Multi-millionaires and billionaires that can throw money and influence around to get what they want and avoid paying taxes they do not want to spend. They are already segregated in part by large mansions with high walls, security guards to enforce their privacy, and the ability to pay more to access things that poorer people cannot.
For the Ruling Class, it is all about money and influence. Their life will revolve around networking, making connections, and expanding family fortunes. Since these are most likely the people that can access the ruler, some will also jockey for influence with the ruler. Others will be the people that administrate the lands.
To keep the lower classes out, the Ruling Class' children will attend an expensive boarding school, with the fees ensuring that they have the best of everything along with a superior experience, and that only they could afford it. This institution will be one part day care, one part school, and one part networking with each class making their life-long connections here. Children will be groomed for their roles, picked for the one they will have the most competency at for the most part. For those that cannot make it for reasons, an unemployment salary of 280,000 dollars (700,000 x 40%) will allow them to be comfortable, though not able to exert much influence.
These will be the people that can afford to own the large corporations and entire apartment buildings that the Working Class live in should they want to own them. They will have servants to attend to them. They will buy up the promising businesses that the Specialist Class start in order to prevent their rise and to keep the profits flowing to them.
The Ruling Class will also control information as since they are the administrators of the realm. It is their job to make sure that people stay in their roles and that people cannot ascend the class ranks. How they do it will depend on what kind of world feel you are going for.
Step 3: Enforced Separation
Now the hard part: enforcing the separation of the classes. Laws brought down by the leader are a start, but there are still a ways to go.
First minor thing: Eliminate the big prize jackpot lotteries/gambling. Dumping multiple millions on a Working Class person allows them to move up, if only virtually because they can afford the Specialist schooling. Once enough get that education because of a lottery win, then the idea of separation starts to break down. Prizes should either have a monetary limit or be non-cash prizes, such as a bungalow or vehicle.
Second: The punishment for intercaste marriage will be a demotion to the lower of the two spouse's castes as dictated by law. The Ruling Class counts on this to be both a deterrent and to provide informants into the lower castes without having to go there themselves when they need to know something.
Third: Jail is for those that cannot pay for their crimes with money and/or influence. This keeps the jails free of the Ruling Class, though their own mansions may serve as a gilded jail in those situations. Sure, it looks like the rich can buy their freedom, but that's just privilege (and it happens now to an extent). Demotion might be the boogeyman's sentence.
Fourth: Ensure that your entertainment is consistent with the reality you want to present. TV shows should not have a Working Class man rising through the classes without either consequences or a clear indication that it is fantasy. The goal is to ensure that the idea that they could realistically move up is never taken seriously.
Lastly: The changes need to be applied subtly or in small increments. Some will need laws to fully enact while others will just be a matter of convention. This is more about nudging things into the directions wanted as opposed to just dropping a new world order on people.